Method for estimation of at least one engine parameter

ABSTRACT

A system comprising determination of tuning parameters enabling optimal estimation of unmeasured engine outputs, e.g., thrust. The level of degradation of engine performance is generally described by unmeasurable health parameters related to each major engine component. Accurate thrust reconstruction depends upon knowledge of these health parameters, but there are usually too few sensors to estimate their values. A set of tuning parameters is determined which accounts for degradation by representing the overall effect of the larger set of health parameters as closely as possible in a least squares sense. The method utilizes the singular value decomposition of a matrix to generate a tuning parameter vector of low enough dimension that it can be estimated by a Kalman filter. Generation of a tuning vector specifically takes into account the variables of interest. The tuning parameters facilitate matching of both measured and unmeasured engine outputs, as well as state variables.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional application of and claims priority ofU.S. Pat. No. 7,860,635 “Singular Value Decomposition-Based Method forOptimal Estimation of Turbofan Engine Thrust and Other UnmeasurableParameters,” by J S Litt, filed May 11, 2007, and claims priority ofU.S. Prov. Appl. No. 60/800,558, by J S Litt, filed May 12, 2006, bothof which are incorporated herein by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention described herein may be manufactured and used by and forthe Government of the United States of America for government purposeswithout the payment of any royalties therefor.

BACKGROUND OF THE INVENTION

I-flight estimation of unmeasurable turbofan engine outputs, such asthrust, is difficult because the values depend on the degradation levelof the engine; which is often not known accurately. Degradation isgenerally defined in terms of off-nominal values of health parameters,such as efficiency and flow capacity, related to each major enginecomponent. By the invention, it is now possible to estimate these healthparameters deviations, given that there are at least as many sensors asparameters to be estimated. In standard engine installations, however,there are typically fewer sensors than health parameters, makingaccurate estimation impossible. An approach used in this situation is toselect a subset of health parameters to estimate, assuming the othersremain unchanged. If any of the unaccounted-for health parametersdeviate from nominal, their effect will be captured to some extent inthe estimated subset. As a result, the estimated values will no longerrepresent the true health parameters deviations. There are examples inthe literature of a subset of health parameter “tuners” being used toreconstruct performance variables such as thrust (Luppold, R. H.,Gallops, G. W., Kerr, L. J., Roman, J. R., 1989, “Estimating I-FlightEngine Performance Variations Using Kalman Filter Concepts,”AIAA-89-2584; Turevskiy, A., Meisner, R., Luppold, R. H., Kern, R. A.,and Fuller, J. W., 2002, “A Model-Based Controller for Commercial AeroGas Turbines,” ASME Paper GT2002-30041; Kobayashi, T., Simon, D. L.,Litt, J. S., 2005, “Application of a Constant Gain Extended KalmanFilter for I-Flight Estimation of Aircraft Engine PerformanceParameters,” ASME Paper GT2005-68494; the entire disclosures of whichare herein incorporated by reference), but this approach of healthparameter subset selection is much better established as a diagnostictool for gas path analysis (Brotherton, T., Volponi A., Luppold, R.,Simon, D. L., 2003, “eSTORM: Enhanced Self Tuning O-board RealtimeEngine Model,” Proceedings of the 2003 IEEE Aerospace Conference;Kobayashi, T., Simon, D. L., 2003, “Application of a Bank of KalmanFilters for Aircraft Engine Fault Diagnostics,” ASME Paper GT2003-38550;Kobayashi, T., Simon, D. L., 2004, “Evaluation of an Enhanced Bank ofKalman Filters for I-Flight Aircraft Engine Sensor Fault Diagnostics,”ASME Paper GT2004-53640; the entire disclosures of which are hereinincorporated by reference), where studies have determined which healthparameters give good indications of certain faults for particular typesof turbine engines (Stamatis, A., Mathioudakis, K., Papailiou, K. D.,1990, “Adaptive Simulation of Gas Turbine Performance,” Journal ofEngineering for Gas Turbines and Power, 112, pp. 168-175; Tsalavoutas,A., Mathioudakis, K., Stamatis, A., Smith, M., 2001, “Identifying Faultsin the Variable Geometry System of a Gas Turbine Compressor,” Journal ofTurbomachinery, 123, pp. 33-39; Ogaji, S. O. T., Sampath, S., Singh, R.,Probert, S. D., 2002, “Parameter Selection for Diagnosing aGas-Turbine's Performance-Deterioration,” Applied Energy, 73, pp. 25-46;the entire disclosures of which are herein incorporated by reference).

When a Kalman filter is used to estimate the subset of healthparameters, the estimates of measured outputs will usually be good,i.e., the sensed outputs and the recreated values obtained using thehealth parameter estimates will match, even if the health parameterestimates themselves are inaccurate. However, good estimation of sensedoutputs does not guarantee that the estimation of unmeasured outputswill be accurate. Since thrust is affected by the level of degradation,poor health parameter estimation can result in poor thrustreconstruction. It might be possible to determine a subset of healthparameters that produces good thrust reconstruction even when all healthparameters deviate, but this is a time-consuming, empirical,trial-and-error process that gives no guarantee about the optimality ofthe result given the potential range of health parameter deviations andoperating conditions.

The main issue that affects the estimation accuracy is that the totalinfluence of the health parameters needs to be approximated using fewervariables. The selection of a subset of health parameters is not ageneral approach to solving this problem as long as all healthparameters may deviate.

BRIEF SUMMARY OF THE INVENTION

Applicant has derived a set of tuning parameters (not necessarily asubset of health parameters) that is smaller in dimension than the setof health parameters, but retains as much information as possible fromthat original set.

The following sections of this application describe and formulate thethrust estimation problem mathematically, and then outline a newapproach to the solution using singular value decomposition to obtain anoptimal set of tuners. Following that, a concise design process isdescribed that generates a set of optimal tuners at each operating(linearization) point, explicitly including all variables of interest;these tuners are incorporated into the Kalman filter to provide theoptimal estimates. The technique is then demonstrated through anexample. After the example, there is further disclosure about thetechnique that covers other applications, such as its use as adiagnostic and control tool.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the state variable estimation over a time sequencecontaining 10 such sets of random shifts.

FIG. 2 shows the output variables for the same time sequence as in FIG.1 of health parameter shifts.

FIG. 3 shows the auxiliary parameters scaled actual and filteredestimates.

FIG. 4 plots the estimated q signals and V*p demonstrating faultdetection.

FIG. 5 is a graphic representation of Kalman filter estimates of healthparameters based turners with the space input sequence as in FIG. 4.

FIG. 6 is a graphic representation of fault signal {circumflex over(p)}≈V*^(T){circumflex over (q)} and actual health parameter shiftdemonstrating fault isolation.

DETAILED DESCRIPTION OF THE INVENTION

In the disclosure of the invention, the nomenclature in the followingtable will be utilized as a shorthand manner of expressing the conceptsutilized in the invention.

NOMENCLATURE A, A_(Aug), B, B_(Aug), C, C_(Aug), System Matrices D, E,E_(Aug), F, G, L, M, N e, w Noise Vectors FAN Fan FG Gross Thrust FN NetThrust HPC High Pressure Compressor HPT High Pressure Turbine LPC LowPressure Compressor LPT Low Pressure Turbine LS Least Squares(subscript) P Vector of health parameters P17 Bypass discharge pressureP25 HPC inlet pressure PS3 Combustor inlet static pressure Q, R Noisecovariance matrices Q Vector of optimal tuners SMW12 Fan stall marginSMW2 LPC stall margin SMW25 HPC stall margin T25 HPC inlet temperatureT3 Combustor inlet temperature T49 LPT inlet temperature TMHS23 LPCmetal temperature TMHS3 HPC metal temperature TMHS41 HPT nozzle metaltemperature TMHS42 HPT metal temperature TMHS5 LPT metal temperatureTMBRNC Combustor case metal temperature TMBRNL Combustor liner metaltemperature U Orthogonal matrix obtained using SVD U* Optimaltransformation matrix u Vector of control inputs u_(i) ith column of U VOrthogonal matrix obtained using SVD V* Optimal transformation matrixv_(i) ith column of V VBV Variable bleed valve VSV Variable stator valveWF36 Fuel flow WR2A Total FAN corrected flow x State vector x_(Aug)Augmented state vector XN12 Low-pressure spool speed XN25 High-pressurespool speed z Vector of auxiliary outputs δ Vector of healthparameter-induced shifts Σ Singular valve matrix obtained using SVDσ_(i) ith singular valveProblem Development

For a linear point designed or a piece-wise linear state variable model,the equations of interest are:X=Ax+Bu+Lp+ey=Cx+Du+Mp+wz=Ex+Fu+Np  (1)where x is the vector of state variables, u is the vector of controlinputs, y is the vector of measured outputs, and z is the vector ofauxiliary (unmeasurable or at least unmeasured) model-based outputs. Thevector p represents the engine health parameters, which induce shifts inother variables as the health parameters move away from their nominalvalues. The vector e represents white process noise with covariance Q,and w represents white measurement noise with covariance R. The matricesA, B, C, D, E, F, L, M, and N are of appropriate dimension.

As the health parameter vector p is an unknown input to the system,applicants are able to estimate it since it affects unmeasurableparameters such as thrust. The health parameters may be treated as a setof biases, and thus are modeled without dynamics. With thisinterpretation, we can represent equation (1) as:

$\begin{matrix}{{\begin{bmatrix}\overset{.}{x} \\\overset{.}{p}\end{bmatrix} = {{{\begin{bmatrix}A & L \\0 & 0\end{bmatrix}\begin{bmatrix}x \\p\end{bmatrix}} + {\begin{bmatrix}B \\0\end{bmatrix}u} + e} = {{A_{Aug}x_{Aug}} + {B_{Aug}u} + e}}}{y = {{{\begin{bmatrix}C & M\end{bmatrix}\begin{bmatrix}x \\p\end{bmatrix}} + {Du} + w} = {{C_{Aug}x_{Aug}} + {Du} + w}}}{z = {{{\begin{bmatrix}E & N\end{bmatrix}\begin{bmatrix}x \\p\end{bmatrix}} + {Fu}} = {{E_{Aug}x_{Aug}} + {{Fu}.}}}}} & (2)\end{matrix}$

This new system has at least as many eigenvalues at the origin as thereare elements of p (the eigenvalues of A_(Aug) are the eigenvalues of Aplus an additional dim(p) zeros due to the augmentation). Once the pvector is appended to the state vector, it may be directly estimated,provided that the realization in equation (2) is observable.

Using this formulation, the number of health parameters that can beestimated is limited to the number of sensors, the dimension of y(Espana, M. D., 1994, “Sensor Biases Effect on the Estimation Algorithmfor Performance-Seeking Controllers,” Journal of Propulsion and Power,10, pp. 527-532). This is easily seen by examining the observabilitycriterion (Callier, F. M., and Desoer, C. A., 1991, Linear SystemsTheory, Springer-Verlag, New York, p. 240; the entire disclosure ofwhich is herein incorporated by reference), which states that forobservability, the matrix [(λI−A_(Aug))^(T)C_(Aug) ^(T)]^(T) must befull rank for each eigenvalue, λ, of A_(Aug), i.e.,

${\forall{\lambda \in \left\{ {{eig}\left( A_{Aug} \right)} \right\}}},{{{rank}\left( \begin{bmatrix}{{\lambda\; I} - A_{Aug}} \\C_{Aug}\end{bmatrix} \right)} = {{\dim\left( x_{Aug} \right)}.}}$

Since the matrix A_(Aug) in equation (2) clearly has at least as manyzero eigenvalues as there are health parameters, the observabilitycriterion matrix reduces to [−A^(T) _(Aug) C^(T) _(Aug)]^(T) for eachzero eigenvalue, implying that for observability, C_(Aug) must have atleast as many rows (i.e., there must be at least as many sensors) asA_(Aug) has rows of zeros.

Since there are usually fewer sensors than health parameters, theproblem becomes one of choosing the best set of tuners for theapplication. This is addressed next.

Problem Formulation

In this application, the objective is to determine a tuning vector oflow enough dimension to be estimated, that represents as much of thehealth parameter information as possible in a known way. This tuningvector should permit shifts in the variables of interest, caused byhealth parameter deviations, to be represented as closely as possible ina least squares sense.

We may define δ as the vector of shifts due to the health parameters,

$\begin{matrix}{\delta = {{\begin{bmatrix}L \\M \\N\end{bmatrix}p} = {{Gp}.}}} & (3)\end{matrix}$

Given that the number of elements in the tuning vector (tuners) is lessthan the number of health parameters, we know that, except for specificcases, the estimated tuners will not represent true health parametervalues. Therefore, there is no reason to model the tuning vector as asubset of existing health parameters and, in fact, δ can be matched moreclosely without this constraint. The key is to find a matrix U* andtuning vector q that correspond to G and p in equation (3), of smallenough dimension that the tuning vector can be estimated, and theproduct approximates δ in a least square sense. That is,δ=Gp≈{circumflex over (δ)}=U*q, pε

, qε

, k<n and J=(δ−{circumflex over (δ)})^(T)(δ−{circumflex over(δ)})={tilde over (δ)}^(T){tilde over (δ)}  (4)is minimized.

For the tuning vector q to contain as much of the information containedin p as possible, p is mapped into q through a transformation V* suchthat q=V*p where V* is full rank. Thus, equation (4) becomes

$\begin{matrix}\begin{matrix}{J = {\left( {\delta - \hat{\delta}} \right)^{T}\left( {\delta - \hat{\delta}} \right)}} \\{= {\left( {G_{p} - {U^{*}q}} \right)^{T}\left( {G_{p} - {U^{*}q}} \right)}} \\{= {{p^{T}\left( {G - {U^{*}V^{*}}} \right)}^{T}\left( {G - {U^{*}V^{*}}} \right){p.}}}\end{matrix} & (5)\end{matrix}$Since G may have rank as large as n (full rank) and the inner dimensionof U*V* is only k, U*V* will not be full rank. Thus, equation (5) may berewritten in vector 2-norm notation,

$\begin{matrix}{J = {\min\limits_{{{rank}{({U^{*}V^{*}})}} = k}{{{\delta - {U^{*}V^{*}p}}}_{2}^{2}.}}} & (6)\end{matrix}$

Another interpretation of equation (5) is that we want to approximate Gby a lower rank matrix through the minimization of the Frobenius norm(the square root of the sum of the squares of each matrix element),i.e.,

$\begin{matrix}{J = {\min\limits_{{{rank}{({U^{*}V^{*}})}} = k}{{{G - {U^{*}V^{*}}}}_{F}.}}} & \;\end{matrix}$

The solution to both of these minimization problems is obtained usingSingular Value Decomposition (SVD) (Stewart, G., W., 1973, Introductionto Matrix Computations, Academic Press, New York, pp. 322-324), and weshall show that SVD leads to an optimal solution of the form U*V*.

Singular Value Decomposition

The Singular Value Decomposition (SVD) of any m×n matrix G, with m≧n,may be defined asG=UΣV ^(T)  (7)where U and V^(T) are orthonormal square matrices, UεR^(m×m), VεR^(n×n),and Σ is a matrix of the same dimensions as G, with the upper portion adiagonal matrix of the singular values of G and the lower portion allzeroes. That

$\Sigma = \begin{bmatrix}\sigma_{1} & 0 & \ldots & 0 \\0 & \sigma_{2} & \; & \vdots \\\vdots & \; & \ddots & 0 \\0 & \ldots & 0 & \sigma_{n} \\0 & \ldots & 0 & 0\end{bmatrix}$where σ₁≧σ₂≧ . . . ≧σ_(n)≧0, UU^(T)=I_(m), and VV^(T)=I_(n).

Some algebraic manipulation shows that equation (7) is equivalent to:

$\begin{matrix}{G = {\sum\limits_{i = 1}^{n}{\sigma_{i}u_{i}v_{i}^{T}}}} & (8)\end{matrix}$where u_(i) is the ith column of U and v₁ is the ith column of V (v^(T)_(i) the ith row of V^(T)). Equation (8) is called the rank onedecomposition of G because G is represented by the sum of rank onematrices.

If the rank of an m×n matrix H is k<n, its SVD is

$\begin{matrix}\begin{matrix}{H = {{\begin{bmatrix}\underset{\underset{U_{k}}{︸}}{\begin{matrix}u_{1} & \ldots & u_{k}\end{matrix}} & u_{k + 1} & \ldots & u_{m}\end{bmatrix}\begin{bmatrix}\Sigma_{k} & 0 \\0 & 0\end{bmatrix}}\begin{bmatrix}\underset{\underset{V_{k}}{︸}}{\begin{matrix}v_{1} & \ldots & v_{k}\end{matrix}} & v_{k + 1} & \ldots & v_{n}\end{bmatrix}}^{T}} \\{= {{{\begin{bmatrix}U_{k} & u_{k + 1} & \ldots & u_{m}\end{bmatrix}\begin{bmatrix}\Sigma_{k} & 0 \\0 & 0\end{bmatrix}}\begin{bmatrix}V_{k} & v_{k + 1} & \ldots & v_{n}\end{bmatrix}}^{T} = {U_{k}\Sigma_{k}V_{k}^{T}}}}\end{matrix} & (9)\end{matrix}$where U_(k) consists of the first k columns of U, V_(k) consists of thefirst k columns of V, and Σ_(k) is the upper left diagonal k×k block ofΣ containing the no-zero singular values.

The linear least squares problem

$\min\limits_{{{rank}{(H)}} = k}{{\delta - {Hp}}}_{2}^{2}$can be solved using equation (9) as

$\begin{matrix}\begin{matrix}{p_{LS} = {H^{\dagger}\delta}} \\{= {{{\begin{bmatrix}V_{k} & v_{k + 1} & \ldots & v_{m}\end{bmatrix}\begin{bmatrix}\Sigma_{k}^{- 1} & 0 \\0 & 0\end{bmatrix}}\begin{bmatrix}U_{k} & u_{k + 1} & \ldots & u_{n}\end{bmatrix}}^{T}\delta}} \\{= {V_{k}\Sigma_{k}^{- 1}U_{k}^{T}\delta}}\end{matrix} & (10)\end{matrix}$where the pseudo-inverse of the no-square, no-full rank, block diagonalsingular value matrix Σ in equation (9) is that matrix's transpose withthe no-zero block inverted.

Additionally, any full rank m×n matrix G, with m≧n, is most closelyapproximated in the Frobenius norm sense by a matrix H of rank k<n,i.e.,

${{{G - H}}_{F} = {\min\limits_{{{rank}{(H)}} = k}{{G - H}}_{F}}},$when H is given by

$\begin{matrix}{{H = {{\sum\limits_{i = 1}^{k}{\sigma_{i}u_{i}v_{i}^{T}}} = {U_{k}\Sigma_{k}V_{k}^{T}}}},} & (11)\end{matrix}$the first k terms of equation (8).

The Frobenius norm is equal to the square root of the sum of the squaresof the singular values of the matrix (which is equal to the square rootof the sum of the squares of each matrix element) so∥G−H∥ _(F)=√{square root over (σ_(k+1) ²+ . . . +σ_(n) ²)}is minimal since the singular values are ordered from largest tosmallest and the difference consists of only the smallest ones.

Thus, equations (9) and (11) are the same, and both imply equation (10).Moreover, the solution has the structure of the original U*V*formulation proposed in equation (5), even though this structure was notassumed in the derivation of these results, it simply falls out as aconsequence of the SVD approach (Stewart, G. W., 1973, Introduction toMatrix Computations, Academic Press, New York, pp. 322-324; the entiredisclosure of which is herein incorporated by reference; or a similartext for derivations, proofs, definitions of norms, etc.).

This demonstrates that using the SVD to obtain the U*V* formulation willindeed give the optimal approximation to δ in a least squares sense. Theprocedure for coming up with the optimal matrices and their use will bedescribed next.

Design Procedure

The design procedure follows directly from the above derivation.

1. For each linearization point, create G by stacking L, M, and N as inequation (3). The L and M matrices must be included for the accuracy ofthe Kalman filter estimates, the variables used to create N are left tothe user's discretion.

2. Compute the magnitude of each row g_(i) of G as g_(i)g_(i) ^(T) forscaling. Premultiply G by

$W = {\begin{bmatrix}\sqrt{g_{1}g_{1}^{T}} & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \sqrt{g_{m}g_{m}^{T}}\end{bmatrix}^{- 1}.}$This is important because there may be several orders of magnitudedifference between the values in different rows of G, and, since the SVDminimizes the difference between the elements of G and U*V* in a leastsquares sense (Frobenius norm), all rows should be about the samemagnitude to prevent the large matrix elements from dominating.Recognizing, however, that some variables need to be estimated moreaccurately than others, additional weights can be incorporated into thediagonal elements of W to account for this relative importance. This isall done after any other scaling or balancing used to develop theinitial realization (eq. 1).3. Compute the SVD as in equation (7) from the scaled G, WG=(WU)ΣV^(T).4. Unscale WU by premultiplying by W⁻¹. This brings the magnitude ofeach row back to its appropriate level and makes it compatible with therest of the realization.5. Generate U* and V* by selecting only the k most significant terms ofequation (8). At most, k will equal the number of sensors less thenumber of zero eigenvalues in the original A matrix. Incorporate the k×ksingular value matrix into U* such that U*=U_(k)Σ_(k) and V*=V_(k) ^(T).Thus

$G = {{\begin{bmatrix}L \\M \\N\end{bmatrix} \approx \hat{G}} = {{\left( {U_{k}\Sigma_{k}} \right)V_{k}^{T}} = {{U^{*}V^{*}} = {{\begin{bmatrix}U_{L}^{*} \\U_{M}^{*} \\U_{N}^{*}\end{bmatrix}V^{*}} = \begin{bmatrix}\hat{L} \\\hat{M} \\\hat{N}\end{bmatrix}}}}}$where U*_(L)V*={circumflex over (L)}≈L, U*_(M)V*={circumflex over(M)}≈M, and U*_(N)V*={circumflex over (N)}≈N. Thus, I have generated acommon right factor for the approximation of all three matrices L, M,and N.6. Using the rows of U* that correspond to the original L, M, and N, setup the Kalman filter using the new augmented system equations

$\begin{matrix}{\begin{bmatrix}\overset{.}{x} \\\overset{.}{q}\end{bmatrix} = {{{\begin{bmatrix}A & U_{L}^{*} \\0 & 0\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {\begin{bmatrix}B \\0\end{bmatrix}u} + {e.y}} = {{{\begin{bmatrix}C & U_{M}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Du} + {w.z}} = {{\begin{bmatrix}E & U_{N}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Fu}}}}} & (12)\end{matrix}$

This system will generally be observable if q is of small enoughdimension.

Turbofan Engine Example

A large commercial turbofan engine model linearized at a cruiseoperating point is used to evaluate the concept. The model has ninestate variables, 10 health parameters, three control inputs, all shownin table 1, and seven sensors, shown in table 2. The auxiliary outputsof interest are shown in table 3. The linear model is used as the“truth” model for this example. It must be noted that for this work, theP25 measurement in the conventional sensor suite is replaced by P17.Without this substitution, thrust estimation is poor. This indicatesthat the ability to estimate unmeasured outputs is strongly influencedby sensor location. The weights on net and gross thrust (step 2 in thedesign procedure) are doubled because of the relative importance ofthese auxiliary variables. The model is run open-loop, so all controlinputs remain at 0, i.e., they do not deviate from the trim value forthe linear model and no actuator bias is present. All 10 healthparameters are shifted randomly up to 3 percent in the direction ofdegradation, and these shifts result in shifts in the measuredvariables. The linear truth model and Kalman filter estimator use thesame A, B, C, and D matrices, but the truth model has all 10 healthparameters entering through the L, M, and N matrices. The Kalman filteris set up using the system in equation (12) with a seven-element qvector.

To demonstrate the optimality of the SVD-based tuner approach, it wascompared to the Kalman filter approach that uses health parameter-basedtuners. To make the comparison as meaningful as possible, the healthparameter tuners were chosen specifically to be compatible with themetric used by the SVD approach (eq. 4). Structurally, the Kalman filteris designed to estimate seven selected health parameter tuners (whileassuming the others remain constant) and the corresponding columns of L,N, and N are retained. An optimal selection process based on orthogonalleast squares (OLS) (Chen, S., Billings, S. A., Luo, W., “OrthogonalLeast Squares Methods and Their Application to No-Linear SystemIdentification,” 1989, International Journal of Control, 50, pp.1873-1896; Chen, S., Cowan, C. F. N., Grant, P. M., 1991, “OrthogonalLeast Squares Learning Algorithm for Radial Basis Function Networks,”IEEE Transactions on Neural Networks, 2, pp. 302-309; the entiredisclosures of which are herein incorporated by reference) is used todetermine which health parameters to estimate. Briefly, the scaled Gmatrix (from step 2 in the design procedure, the same one as used forthe SVD-based tuners) is multiplied by a random health parameter vectorp whose 10 elements take on values from 0 to 3 percent in theappropriate direction (positive for the HPT and LPT flow capacities,negative for all others). OLS is used to choose the seven columns of Gthat best align with the shift vector δ=Gp. This is repeated thousandsof times for different random p vectors and the results are tallied. Theindividual columns that are selected most often determine the healthparameters that on average best capture the degradation-induced shifts.The most often selected columns are indicated by asterisk (*) intable 1. Of the 120 (10 choose 7) possible sets of columns, this exactset was selected about 9 percent of the time.

TABLE 1 STATE VARIABLES, HEALTH PARAMETERS, AND ACTUATORS StateVariables Health Parameters Actuators XN12 FAN efficiency^(‡) WF36 XN25FAN flow capacity*^(‡§) VBV TMHS23 LPC efficiency*^(‡) VSV TMHS3 LPCflow capacity^(‡§) TMBRNL HPC efficiency*^(‡) TMBRNC HPC flowcapacity*^(‡§) TMHS41 HPT efficiency*^(‡§) TMHS42 HPT flowcapacity*^(‡§) TMHS5 LPT efficiency LPT flow capacity*^(§)

TABLE 2 SENSOR SETS AND SENSOR STANDARD DEVIATION (STD. DEV. AS % OFSTEADY-STATE VALUES AT FULL POWER Conventional New Standard DeviationSensor Set Sensor Set (%) XN12 XN12 0.25 XN25 XN25 0.25 P17 0.50 P250.50 T25 T25 0.75 PS3 PS3 0.50 T3 T3 0.75 T49 T49 0.75

TABLE 3 AUXILIARY OUTPUTS TO BE ESTIMATED IN FLIGHT Auxiliary OutputsWR2A FN FG SMW12 SMW2 SMW25Results

To demonstrate the ability of the optimal tuners to approximate avariety of health parameter-induced shifts well, all 10 healthparameters are shifted randomly. The results shown are for arepresentative random sequence, many simultaneous random shifts wereperformed at this operating point with similar results. FIG. 1 shows thestate variable estimation over a time sequence containing 10 such setsof random shifts. Generally, the variables are reconstructed accuratelywith the exception of TMBRNC, combustor case metal temperature, whichdoes not seem to have much of an effect on any other variables.

FIG. 2 shows the output variables for the same time sequence as in FIG.1 of health parameter shifts. These also demonstrate good matching,which is to be expected when using a Kalman filter.

FIG. 3 shows the auxiliary parameters scaled actual and filteredestimates. For all case of health parameter shifts, both estimatorsperformed well. Because of the large gains on some of the auxiliaryvariables, the estimates needed to be low-pass filtered to attenuate themagnified noise level. The estimation performance on net and grossthrust and two of the three stall margins, as well as total fancorrected flow was consistently accurate. The only auxiliary parameterwhose estimate was consistently poor is LPC stall margin, SMW2.Adjustment of the weighting matrix W in step 2 of the design proceduremade little difference in the results.

In an effort to obtain a better LPC stall margin estimate, new U* and V*were determined, this time using SMW2 as the only auxiliary parameter,i.e., N in equation (3) has only one row. If SMW2 requires differentinformation than the other variables for accurate estimation, generatinga new G matrix would result in a different SVD (the SVD of a matrix isunique) possibly prioritizing the tuners such that SMW2 could bereconstructed. Even with the new G matrix though, the estimation of SMW2is poor. This may be due to the L and M matrices dominating thegeneration of the U* and V* matrices, but it is more likely a sensorplacement issue (in fact, replacing P17 with P25 greatly improves theestimation of SMW2, even without additional weighting).

Likewise, a slight improvement in net and gross thrust estimation wasachieved by the SVD-based tuners when the auxiliary parameters consistedof only those two variables.

In order to compare the performance of the tuners statistically, runscontaining a sequence of 400 simultaneous random shifts of all 10 healthparameters were performed. After each set of shifts, the enginesimulation was allowed to reach steady state and data from each variablewere gathered and averaged. This information was used to compute anaverage percent error for each variable in steady state over the entiresequence. The quantity was computed as:

$\begin{matrix}{{\%\mspace{14mu}{error}} = {\frac{100}{400}{\sum\limits_{i = 1}^{400}{\frac{{{actual}\mspace{14mu}{deviation}_{i}} - {{estimated}\mspace{14mu}{deviation}_{i}}}{{trim}\mspace{14mu}{value}_{i}}}}}} \\{\approx {\frac{100}{400}{\sum\limits_{i = 1}^{400}{\frac{{actual}_{i} - {estimate}_{i}}{{actual}_{i}}}}}}\end{matrix}$

The comparison of the percent estimation error of the state variables,outputs, and auxiliary outputs based on this run for the two sets oftuners are presented in table 4, table 5, and table 6, respectively. Itcan be seen that both sets of tuners produce similar results in mostcases, but where the error is relatively large (TMBNC in table 4 andSMW2 and SMW25 in table 6), the SVD-based tuners are significantlybetter. Results for the estimated degradation-induced shifts in eachvariable, δ=Gp (eq. 3), were similar. The value of the metric (eq. 4)for the scaled shifts in the two cases was 2.187 for the SVD-basedtuners and 6.016 for the health parameter-based tuners. It must be notedthat minimization of the metric does not in itself guaranteeminimization of the total least squares error without approximateweighting in step 2 of the design procedure. In this example, theSVD-based tuners give a value of 42.6865 for the sum of the squaredpercent estimation error of the variables in table 4, table 5, and table6, compared to 95.4661 for the health parameter-based tuners.

TABLE 4 PERCENT ERROR OF STATE VARIABLE ESTIMATES Health Parameter-State Variables SVD-based Tuners Based Tuners XN12 0.0318 0.0317 XN250.0269 0.0267 TMHS23 0.0129 0.0130 TMHS3 0.0363 0.0363 TMBRNL 0.02780.0306 TMBRNC 0.4083 1.1256 TMHS41 0.0300 0.0313 TMHS42 0.0267 0.0274TMHS5 0.3208 0.1942

TABLE 5 PERCENT ERROR OF OUTPUT ESTIMATES Health Parameter- OutputsSVD-based Tuners Based Tuners XN12 0.0317 0.0318 XN25 0.0286 0.0285 P170.0634 0.0633 T25 0.0859 0.0857 PS3 0.1027 0.1027 T3 0.0990 0.0990 T490.0748 0.0748

TABLE 6 PERCENT ERROR OF AUXILIARY OUTPUT ESTIMATES Health Parameter-Auxiliary Outputs SVD-based tuners based Tuners WR2A 0.1081 0.0787 FN0.2704 0.2745 FG 0.1225 0.1169 SMW12 0.4348 0.3469 SMW2 6.1605 9.3142SMW25 2.0326 2.6733

Additionally, estimation performance was evaluated for comparablesequences of 400 random simultaneous shifts in all 10 health parameterswith six sensors and with eight sensors. For the eight-sensor case, P25was added to the “new sensor set” of seven shown in table 2. The tunersused were the first eight health parameters, indicated by a (‡) in table1, (LPT efficiency and flow capacity were not used), selected using OLS.As expected, the estimate of SMW2, which was poor with only sevensensors, was much more accurate, as was the estimate of TMBRNC. Foreight sensors, the value of the metric for the SVD-based tuners was2.0883 and 6.5243 for the health parameter-based tuners; this isessentially the same as for the seven sensor case, with the differencebeing attributed to the random factors of sensor noise and the healthparameter shifts. The sum of the squared percent estimation error forthe SVD-based tuners was 6.0438 compared to 7.1943 for the healthparameter-based tuners. For the six sensor case, the T3 sensor wasremoved from the “new sensor set” shown in table 2. The tuners used werethe flow capacities of each component plus HPT efficiency, indicated by(§) in table 1, which were selected using engineering judgment. It waspossible to get reasonably good matching of the auxiliary outputs withthis set of tuners, but no set investigated produced accuratereconstruction of the state variables. The value of the metric for theSVD-based tuners in the six-sensor case was 9.4068 and 12.4667 for thehealth parameter-based tuners. The big increase in the metric valuesover the cases with more sensors indicates that significant informationhas been lost by approximating the G matrix by a rank six matrix. Therelative size of the dropped singular values gives an indication ofthis. In this case, the sum of the squared percent estimation error forthe SVD-based tuners was 53.8702 compared to 78.1314 for the healthparameter-based tuners.

Further Application of Tuners

The example shows that the optimal tuner selection method obtained usingSVD provides accurate estimates of most variables of interest, includingauxiliary parameters such as thrust. The SVD formulation has severalattractive properties that make its use in this application highlydesirable beyond what has already been discussed. In addition toproviding the least squares solution to the approximation of the shiftvector δ, other features include 1) the approximation error to thematrix G is known; 2) fault detection is possible through the tuners'ability to capture sudden shifts, even if the tuners do not correspondto individual health parameters; and 3) the orthogonality of the Vmatrix suggests a way to isolate a component fault, while at the sametime some of the true health parameter shifts may be able to bereconstructed accurately, in some cases. The following sections providemore detail on each of these topics.

Approximation Error of G

Since

$\begin{matrix}{G = {{\sum\limits_{i = 1}^{n}{\sigma_{i}u_{i}v_{i}^{T}}} = {{{\sum\limits_{i = 1}^{k}{\sigma_{i}u_{i}v_{i}^{T}}} + {\sum\limits_{i = {k + 1}}^{n}{\sigma_{i}u_{i}v_{i}^{T}}}} = {\hat{G} + {\sum\limits_{i = {k + 1}}^{n}{\sigma_{i}u_{i}v_{i}^{T}}}}}}} & (13)\end{matrix}$the approximation error of G is the last term of equation (13), namely.

$\overset{\sim}{G} = {{G - \hat{G}} = {\sum\limits_{i = {k + 1}}^{n}{\sigma_{i}u_{i}{v_{i}^{T}.}}}}$

The error in the approximation of G depends on both the number of termsdropped in the approximation and on the size of the singular value ineach dropped term. (Singular values given an indication of the linearindependence of the columns of G, so if the singular values in thedropped terms are very small, not much information is lost, but on theother hand, if the columns of G are not linearly independent, theaccurate estimation of the health parameters is impossible, regardlessof the number of sensors.) The fact that the error is known allows, forinstance, the calculation of accuracy bounds on the healthparameter-induced shifts, δ, and on the estimated variables themselves.

The error introduced into the system equations (eq. 12) through theapproximation of G represents modeling error. The process noise term, e,in the state equation, is used at least partially to represent modeluncertainty, so a level can be established by multiplying the knownapproximation error {tilde over (G)} by a range of degradation values p.Since the level of degradation is unknown a priori, it can be considereda random variable and, thus, so can {tilde over (G)} p. The covarianceof this product could be used to help determine the process noisecovariance matrix, Q, for the Kalman filter, which is often considered adesign parameter.

Fault Detection

Even though the tuners do not represent individual health parameters,component faults can be detected by sudden shifts. The objective offault detection in this context is to be able to distinguish an abruptshift representing a fault, from the general “background level” ofdegradation caused by use and wear, represented by slowly varying qduring normal operation. Since q=V*p, the shifts are mapped into q aslong as they are not limited to the null space of V*, that is, as longas they do not lie completely within the span of [v_(k+1) . . .v_(n)]^(T). However, since this space contains the least significantdirections of possible shift (since only the smallest singular valuesare dropped) and the tuners will, in general, cut across rather thanalign with the influence of each health parameter, additional shifts dueto faults Δq=V*Δp will very likely be no-zero for any significant

To demonstrate this, component faults of 3 percent magnitude wereinjected, sustained for 20 seconds, and removed, in succession througheach of the 10 health parameters (efficiency then flow capacity for eachcomponent in the order shown in table 1), as shown in table 7.

TABLE 7 HEALTH PARAMETER SHIFTS REPRESENTING COMPONENT FAULTS Time (sec)Health Parameters Shift  0-20 FAN efficiency −3% 20-40 FAN flow capacity−3% 40-60 LPC efficiency −3% 60-80 LPC flow capacity −3%  80-100 HPCefficiency −3% 100-120 HPC flow capacity −3% 120-140 HPT efficiency −3%140-160 HPT flow capacity  3% 160-180 LPT efficiency −3% 180-200 LPTflow capacity  3%

FIG. 4 shows the estimated tuners during this sequence of shifts. As thehealth parameters shift the tuners generally shift as well, so changescan be determined and thus faults can be detected. The fan efficiencyfault is hard to determine as the tuner shifts that correspond to it aresmall, but several tuners clearly shift in response to a fault in fanflow capacity. When looking at all tuners together, it is reasonablyclear in most cases when a fault has occurred. Different sets of tunersresult from different weightings of the G matrix (step 2 of the designprocedure), and it is possible to have at least one of the SVD-basedtuners align well with an actual health parameter.

One thing that is clear from FIG. 4 is that in this example, theelements of q are tracked accurately, and this is a consequence of theprocedure used to select them. This is not the case in general when thetuners are modeled as actual health parameters. FIG. 5 shows the Kalmanfilter estimates of the health parameter-based tuners with the sameinput sequence as in FIG. 4. In FIG. 5, the tuners clearly indicateabrupt shifts due to faults, but often do not match the healthparameters they represent because of the influence of unmodeled healthparameters. Techniques exist to select the health parameters forestimation that are least affected by the unmodeled health parametersand sensor uncertainty (Stamatis, A., Mathioudakis, K., Papailiou, K.D., 1992, “Optimal Measurement and Health Index Selection for GasTurbine Performance Status and Fault Diagnosis,” Journal of Engineeringfor Gas Turbines and Power, 114, pp. 209-216; Gronstedt, T. U. J., 2002,“Identifiability in Multi-Point Gas Turbine Parameter EstimationProblems,” ASME Paper GT2002-30020; Pinelli, M., Sina, P. R., 2002, “GasTurbine Field Performance Determination: Sources of Uncertainties,”Journal of Engineering for Gas Turbines and Power, 124, pp. 155-160;Mathioudakis, K., Kamboukos, Ph., 2004, “Assessment of the Effectivenessof Gas Path Diagnostic Schemes,” ASME Paper GT2004-53862; the entiredisclosures of which are herein incorporated by reference). From adiagnostic point of view, this is fine, but there is no guarantee thatthose health parameters will facilitate good reconstruction of auxiliaryparameters such as thrust.

The ability to estimate the state and auxiliary variables in thepresence of actuator bias is not addressed here. From equation (1) it isclear that the effect of actuator bias is no different than that ofhealth parameter shifts, so as long as the tuners can capture thiseffect, the results should not deteriorate significantly. As withcomponent degradation and faults, constant actuators biases should begenerally represented by the elements of q, and any sudden actuatorshifts due to bias should be captured as shifts in the elements of q.

Fault Isolation

By construction, the vector of estimated tuners q≈V*p, (since G is notperfectly recreated by U*V*, the estimated q will not exactly equal V*p,even in a noise-free environment) and the columns of V are orthornormal.Of course, V is a square n×n matrix and V* is k×n, but if k is close ton, in some cases V*^(T)V*≈I_(n). The implication of this is thatV*^(T)q≈p, so even if the magnitude of the shift is not determinedaccurately, the shift might be able to be isolated to the affectedhealth parameters (or at least the affected component) by identifyingthose elements of the V*^(T)q vector with the largest shifts. Analysisof the V* matrix will determine the feasibility of the approach on acase by case basis.

It is interesting to note that the V*^(T)q≈p approximation is actuallythe least squares estimate of p from equation (3). This is clear fromthe fact that:{circumflex over (δ)}=U*{circumflex over (q)}=U _(k)Σ_(k) {circumflexover (q)}≈Gp=UΣV ^(T) p

p _(LS) =VΣ ^(†) U ⁻¹ U _(k)Σ_(k) {circumflex over (q)}=V _(k){circumflex over (q)}=V* ^(T) {circumflex over (q)}  (14)where Σ^(†) is the transpose of Σ with the square singular value blockinverted. Note that once U is unscaled in step 4 of the designprocedure, it is no longer orthonormal, so U⁻¹ rather than U^(T) is usedin equation (14).

FIG. 6 shows the recreated set of health parameters using p≈V*^(T)q. Itis clear that shifts in certain variables are easily isolated whileothers are mapped back as shifts smeared across more than one healthparameter. For this example, it seems that health parameter shifts inthe high-pressure shaft components (HPC and HPT) are easier to isolatethan those on the low-pressure shaft. For instance, the plots for FANefficiency and LPT efficiency are very similar, meaning that it would behard for a diagnostic system to distinguish between them. Efficiency andflow capacity of the LPC are clearly interrelated, with one being nearlythe opposite of the other; for cases such as this, one might constrainthe problem by assuming that efficiency and flow capacity correspondingto a single component shift together, somewhat simplifying the isolationlogic. The issue always exists of simultaneous shifts in the healthparameters of multiple components confounding the problem.

It is interesting to compare the results in FIG. 6 with those in FIG. 5,which shows the Kalman filter estimates of seven health parameter-basedtuners. The quality of the reconstruction is similar in that the samethree parameters (FAN flow capacity, HPC efficiency, and HPT flowcapacity) were estimated accurately in both cases and those estimatescorrupted by unmodeled shifts in health parameters were the same in bothcases, but the approach demonstrated in FIG. 6 has the benefit that theremaining three health parameters are reconstructed as well, albeitcorrupted by shifts in other variables. One observation is that thereconstruction of each of these remaining health parameters (FANefficiency, LPC flow capacity, and LPT efficiency) is strongly relatedto the reconstruction of some other health parameter, underscoring thelack of observability. Still, a diagnostic system might be able to usethis information to isolate faults.

If actuator bias, u_(bias) were to be added on top of the healthparameter deviations, it would certainly make the fault isolation taskmore challenging. An approach to address this is to define G such that:

$\delta = {{\begin{bmatrix}L & B \\M & D \\N & F\end{bmatrix}\begin{bmatrix}p \\u_{bias}\end{bmatrix}} = {{G\begin{bmatrix}p \\u_{bias}\end{bmatrix}}.}}$This explicitly accounts for actuator bias, but the increase in thenumber of columns of G means that the U*V* approximation will be lessaccurate, since the size of q is fixed. A consequence of this is thatV*^(T)V* is further from I_(n) (in the sense of the Frobenius norm) and,thus, the [p^(T) u_(bias) ^(T)]^(T)≈V*^(T)q approximation is not as goodas when fewer faults are accounted for.Commercial Benefits

A new optimal linear point design approach to determine enginedeterioration tuning parameters that enables the accurate reconstructionof unmeasured engine outputs is presented. It was designed specificallyfor the case where there are too few sensors to estimate the true enginehealth parameters. The tuning parameters are determined using singularvalue decomposition, which was shown to generate the best approximationto the influence of the full set of health parameters in a least squaressense, using a set small enough to be estimated. A concise designprocedure to generate the tuners is described. An example demonstratedthat the method worked well in reconstructing the unmeasured parametersof interest. It was shown to perform better against its metric than ahealth parameter-based Kalman filter specifically designed for themetric used, and it will generally result in a smaller total squaredestimation error with appropriate weighting incorporated into themetric. This highlights the freedom inherent in the SVD-based approachas a result of the weighting, while the selection of possible subsets ofhealth parameters for tuners is limited. Still, the choice of weights inthe SVD approach is not arbitrary since the variables interact, andallowing poor state estimation, for instance, will ultimately hurt theaccuracy of the auxiliary variables. It was also shown that the abilityto estimate well is sensitive to sensor placement. As a side benefit ofthe estimator formulation process, the tuners have desirable propertiesfor diagnostics because of the orthogonal decomposition. In particular,the reconstruction of the complete set of health parameters comparesfavorably to the estimation of only a subset by a health parameter-basedKalman filter. The effect of actuator bias was not investigated withrespect to the optimal tuners determined for health parameterdegradation, however, a consistent approach to account for it explicitlyis proposed.

The invention may be used to detect faults in unmeasured parameters andmay be used to control an i-flight engine by employing a computer todetermine the relationship of the tuning factor q to unmeasuredparameters and estimate unmeasured auxiliary parameters of the enginethat are affected by unmeasured health parameters p.

Although I have described the invention with regard to certainembodiments thereof, the invention is not so limited and those ofordinary skill in the art, reading the foregoing specification andappended drawings, will be able to employ or modify the same withoutdeparting from the scope of the following claims.

1. A method of measuring the health of an engine using h number ofparameters, where h is defined as a complete set of health parameterssufficient to fully define a health condition, said method comprising:measuring a predetermined number of engine outputs using sensors, thepredetermined number being less than h; estimating unmeasured outputs ofan engine using a tuning vector q in combination with a transformationmatrix V* where q=V*p, where p is a vector of health parameters h, and qcontains some information from each of the h number of parameters neededto compute the health condition; and estimating the health condition ofsaid engine based upon measured engine outputs.
 2. The method of claim1, wherein the engine is a turbofan engine and the unmeasured parameterswhich are estimated by the tuning vector q in conjunction with atransformation matrix are capable of detecting a stall condition duringthe flight of an aircraft.
 3. The method of claim 1, wherein theunmeasured output that is estimated using the tuning vector is oneselected from the group consisting of gross thrust and net thrust. 4.The method of claim 1, wherein the step of estimating the unmeasuredoutputs comprises determining degraded operation of an engine due toshifts in the health parameters, and wherein the health parametersinclude fan efficiency, fan flow capacity, low pressure compressorefficiency, low pressure compressor flow capacity; high pressurecompressor efficiency; high pressure compressor flow capacity; highpressure turbine efficiency; high pressure turbine flow capacity; lowpressure turbine efficiency and low pressure turbine flow capacity. 5.The method of claim 1, wherein the tuning vector is represented by q andits estimate of value corresponds to the relationship expressed as:$\begin{bmatrix}\overset{.}{x} \\\overset{.}{q}\end{bmatrix} = {{\begin{bmatrix}A & U_{L}^{*} \\0 & 0\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {\begin{bmatrix}B \\0\end{bmatrix}u} + e}$ $y = {{\begin{bmatrix}C & U_{M}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Du} + w}$ $z = {{\begin{bmatrix}E & U_{N}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Fu}}$ where x is the vector of state variables, y isthe vector of measured outputs, z is a vector comprising unmeasuredmodel-based outputs, x(dot) is the derivative dx/dt, q is a tuningvector, q(dot) is the derivative dq/dt, U is an orthogonal square matrixderived from the Singular Value Decomposition of a matrix that mapsengine health parameters into x, y, and z, k is equal to the number ofsensors less the number of zero eigenvalues of the matrix A, U*_(L)consists of the first k columns and the first l rows of U, where lequals the number of state variables x, U*_(M) consists of the first kcolumns and the rows from l+1 to l+m of U, where m equals the number ofmeasured outputs y, U*_(N) consists of the first k columns and rowsl+m+1 to l+m+n of U, where n equals the number of unmeasured outputs z,w represents white measurement noise, e represents white process noise,and A, B, C, D, E, and F are matrices based upon the linear enginemodel.
 6. The method of claim 5, wherein the measured parameters aremeasured by sensors y, and further selecting a k number of values, wherek is equal to the number of sensors less the number of zero eigenvaluesin the matrix A.
 7. A method of determining faults in a turbofan enginewhere the fault is not in a measured parameter, by estimating the tuningvector q of claim 5 a number of times and determining abrupt shifts inthe tuner as an indication of a fault.
 8. The method of claim 1 whereinp is mapped into q through the transformation matrix V* such that q=V*pwhere V* is full rank.
 9. The method of claim 1 wherein the step ofestimating unmeasured engine outputs using a tuning vector comprisesemploying a computer to estimate the tuning vector q a number of timesand determining abrupt shifts in the tuning vector as an indication of afault.
 10. The method of claim 9, where the unmeasured parameter is oneselected from the group consisting of gross thrust and net thrust. 11.The method of claim 9 wherein the unmeasured parameters comprises netthrust and gross thrust the method is used to control an i-flightengine.
 12. The method of claim 11, wherein the relationship betweentuning vector q and the unmeasured parameters satisfies the expression$\begin{bmatrix}\overset{.}{x} \\\overset{.}{q}\end{bmatrix} = {{\begin{bmatrix}A & U_{L}^{*} \\0 & 0\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {\begin{bmatrix}B \\0\end{bmatrix}u} + e}$ $y = {{\begin{bmatrix}C & U_{M}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Du} + w}$ $z = {{\begin{bmatrix}E & U_{N}^{*}\end{bmatrix}\begin{bmatrix}x \\q\end{bmatrix}} + {Fu}}$ where x is the vector of state variables, y isthe vector of measured outputs, z is a vector comprising unmeasuredmodel-based outputs, x(dot) is the derivative dx/dt, q is a tuningvector, q(dot) is the derivative dq/dt, U is an orthogonal square matrixderived from the Singular Value Decomposition of a matrix that mapsengine health parameters into x, y, and z, k is equal to the number ofsensors less the number of zero eigenvalues of the matrix A, U*_(L)consists of the first k columns and the first l rows of U, where lequals the number of state variables x, U*_(M) consists of the first kcolumns and the rows from l+1 to l+m of U, where m equals the number ofmeasured outputs y, U*_(N) consists of the first k columns and rowsl+m+1 to l+m+n of U, where n equals the number of unmeasured outputs z,w represents white measurement noise, e represents white process noise,and A, B, C, D, E, and F are matrices based upon the linear enginemodel.
 13. The method of claim 1 wherein engine component performance isestimated by health parameters in the following state-space equations{dot over (x)}=Ax+Lh+Bu _(cmd)y=Cx+Mh+Du _(cmd) +v where the vector x represents the state variables,x(dot) is the derivative dx/dt, h represents health parameters, u_(cmd)represents control variables, y represents the sensor measurement vectorwhich is corrupted by the noise vector v; and the matrices A, B, C, D, Land M have appropriate dimensions.
 14. The method of claim 1 wherein thehealth parameters define a health condition sufficient to determinedeterioration manifested as a shift in operating point from a baselineengine.